Proving a function is well defined and continuous

[mick25]

Homework Statement

Let $f_{n}(x)=\frac{-x^2+2x-2x/n+n-1+2/n-1/n^2}{(n ln(n))^2}$

Prove $f(x) = \sum^{\infty}_{n=1} f_{n}(x)$ is well defined and continuous on the interval [0,1].

Homework Equations

In a complete normed space, if $\sum x_{k}$converges absolutely, then it converges.

The Attempt at a Solution

Working in a complete normed space $(C[0,1], || . ||_{∞})$,

consider the real series $\sum^{∞}_{n=1}||f_{n}||_{∞}=\sum^{∞}_{n=1} sup <f_{n}(x) : x\in[0,1]>$

It just remains to show that $\sum^{∞}_{n=1}|f_{n}|$ converges, but I can't seem to figure out how. Could anyone help me out here?

 

[lanedance]

some ideas i haven't tried them yet though... how about first separating n into some manageable pieces...

first evaluate the convergence of the terms only dependent on n, then consider the terms with an x and have a think about what x value will maximize that sum.

then if it's tough to pull it together you may want to consider some properties of the norm.

then I think you may need more for the continuous part and move to the epsilon deltas

 

[mick25]

Rather than dealing with all the terms separately, I've been trying to just apply a simple comparison test (followed by an integral test) to prove its convergence, but I could only come up with divergent cases.

I'm starting to think if there is an error with this question; does this series converge if it starts at n=1?

 

[lanedance]

yeah as 1/ln(1)^2 is undefined